The twins are the fifth and sixth children for Jolie, 33, and partner Brad Pitt, 44, who are already parents to Maddox, 6; Pax, 4; Zahara, 3; and Shiloh, 2.

"The mother and father are very, very happy," Sussmann told PEOPLE.

The doctor told the Associated Press that the Cesarean was moved forward "for medical reasons" so that the babies could be born "in the best conditions." He also said Jolie is expected to stay in the hospital for a few days.

On July 1, Jolie checked in to the Fondation Lenval hospital to await the births. The family has been living in the south of France since this spring.

Jolie recently told Entertainment Weekly that she was surprised – but not daunted – when she learned she was having twins. "It did shock us, and we jumped to six [children] quickly," she said. "But we like a challenge."

Celebrity Babies: Jennifer Lopez, Nicole Richie, Christina Aguilera and more show off their adorable additions on the cover of PEOPLE